Pure Mathematics No. 29 ISSN 0806–2439 October 2004
CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF INTEGRO-PDES
ESPEN R. JAKOBSEN AND KENNETH H. KARLSEN
Abstract. We present a general framework for deriving continuous depen- dence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to pro- vide explicit estimates for the continuous dependence on the coefficients and the “L´evy measure” in the Bellman/Isaacs integro-PDEs arising in stochas- tic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock mar- ket driven by a geometric L´evy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods.
Contents
1. Introduction 2
2. Viscosity solution theory for integro-PDEs. 5
3. Continuous Dependence Estimates. 8
4. The Bellman/Isaacs equation 14
4.1. The obstacle problem 20
5. Applications 21
5.1. Regularity of solutions. 21
5.2. The vanishing viscosity method 22
5.3. The vanishing jump viscosity method 23
5.4. A singular perturbation problem by J.-L. Lions and S. Koike. 24 6. Continuous dependence in the Black-Scholes model 25
6.1. Introduction 25
6.2. Option pricing in L´evy markets 25
6.3. Integro-PDEs and continuous dependence 26
6.4. Examples of L´evy models 29
References 30
Date: October 7, 2004.
Key words and phrases. nonlinear degenerate parabolic integro-partial differential equation, Bellman equation, Isaacs equation, viscosity solution, continuous dependence estimate, regularity, vanishing viscosity method, convergence rate.
Jakobsen was partially supported by the Research Council of Norway, grant no. 151608/432.
Karlsen was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. Parts of this work were done while E. R. Jakobsen visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.
1
1. Introduction
The theory of viscosity solutions for fully nonlinear degenerate elliptic/parabolic PDEs is now highly developed [4, 5, 18, 25]. In recent years we have witnessed an interest in extending viscosity solution theory to integro-PDEs [1, 2, 3, 6, 11, 12, 13, 35, 41, 43, 45, 46, 48, 49]. Such non-local equations occur in the theory of optimal control of jump-diffusion (L´evy) processes and find many applications in mathematical finance, see, e.g., [1, 2, 3, 11, 12, 13, 27] and the references cited therein. We refer to the books [28, 29] for an investigation of integro-PDEs by completely different methods.
In this paper we are interested in “continuous dependence on the nonlinearities”
estimates and various consequences of such estimates for viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. To be as general as possible, we write these equations in the form
ut(t, x) +F(t, x, u(t, x), Du(t, x), D2u(t, x), u(t,·)) = 0 in QT, u(0, x) =u0(x) in RN, (1.1)
where QT := (0, T)×RN andF : QT ×R×RN ×SN ×Cp2(RN)→Ris a given functional. HereSN denotes the space of symmetricN ×N real valued matrices, and Cp2(RN) denotes the space of C2(RN) functions with polynomial growth of orderp≥0 at infinity.
These equations are non-local as is indicated by the u(t,·)-term in (1.1). A simple example of such an equation is
(1.2) ut−
Z
RM\{0}
[u(·,·+z)−u−zDu]π(dz) = 0 in QT,
whereπ(dz) is a positive Radon measure onRM\ {0}(the so-called L´evy measure) with a singularity at the origin satisfying
(1.3)
Z
RM\{0}
|z|21B(0,1)+|z|p1B(0,1)c
π(dz)<∞.
Note that the L´evy measure integrates functions withp-th order polynomial growth at infinity. In view of (1.3) and a Taylor expansion of the integrand, the integro operator in (1.2) is well defined onCp2(RN). Moreover, it is clear that the integro operator in (1.2) acts as a non-local second order term, and for that reason the
“order” of the integro operator is said to be two. If |z|2 in (1.3) is replaced by
|z|, this changes the order of the integro operator from two to one, since then it acts just like a non-local first order term. Finally, if|z|2 in (1.3) is replaced by 1 (i.e.,π(dz) is a bounded measure), then the integro operator in (1.2) is said to be bounded or of order zero, and in this case the integro operator acts like a non-local zeroth order term.
An important example of a non-local equation of the form (1.1) is the non-convex Isaacs equations associated with zero-sum, two-player stochastic differential games (see, e.g., [26] for the casewithout jumps)
ut+ inf
α∈Asup
β∈B
−Lα,βu− Bα,βu+fα,β = 0 in QT, (1.4)
whereAandBare compact metric spaces and for any sufficiently regularφ
(1.5)
Lα,βφ(t, x) = tr
aα,β(t, x)D2φ
+bα,β(t, x)Dφ−cα,β(t, x)φ, aα,β(t, x) = 1
2σα,β(t, x)σα,β T(t, x)≥0, Bα,βφ(t, x) =
Z
RM\{0}
φ(t, x+jα,β(t, x, z))−φ−jα,β(t, x, z)Dφ π(dz).
Here tr andT denote the trace and transpose of matrices. The L´evy measureπ(dz) is a positive Radon measure on RM \ {0}, M ≥ 1, satisfying a condition similar to (1.3), see (A0) and (A4) in Section 4. Also see Section 4 for the (standard) regularity assumptions on the coefficients, σ, b, c, andη. We remark that ifA is a singleton, then equation (1.4) becomes the convex Bellman equation associated with optimal control of L´evy (jump-diffusion) processes over a finite horizon (see, e.g., [41, 43] and the references therein). Henceforth we will refer to (1.4) simply as the “Bellman/Isaacs equation”.
The general problem we are confronted with here is to find an upper bound on the difference between a viscosity subsolution uof (1.1) and a viscosity supersolution
¯
uof (1.1) with F replaced by another nonlinear functional ¯F satisfying the same assumptions asF. The sought upper bound foru−¯ushould be expressed in terms of
“F−F¯”. Let us give an explicit example of the type of results that can be obtained with our general continuous dependence framework for integro-PDEs (1.1). Letu be a viscosity subsolution of (1.2) and let ¯ube a viscosity supersolution of
(1.6) u¯t− Z
RM\{0}
[¯u(·,·+z)−u¯−zD¯u] ¯π(dz) = 0 in QT,
where ¯π(dz) is another L´evy measure satisfying (1.3). For simplicity, suppose that the viscosity sub- and supersolutions are bounded, the initial values are zero, and that the L´evy measures admit densities (which is the typical case in finance appli- cations, see Section 6), i.e.,
π(dz) =m(z)dz, π(dz) = ¯¯ m(z)dz,
for some functions m(z) and ¯m(z) that may have singularities at the origin. Our continuous dependence result then yields for any (t, x)∈QT
(u−u)(t, x)¯ ≤C s
T Z
RN\{0}
|z|2|(m−m)(z)|¯ dz.
(1.7)
In other words, the difference betweenuand ¯uis expressed in terms of a weighted L1 norm of the difference between the L´evy densities m and ¯m. Note that it is important that the L1 norm is weighted with the function |z|2, as the densities may have singularities at the origin. The reason for the “square-root” is that the estimate is robust with respect to the smoothness ofuand ¯u. Ifuand ¯uare both viscosity solutions, then, by reversing the roles of uand ¯u, we obtain an estimate for|u−u|. Results similar to (1.7) will be stated for the Bellman/Isaacs equation¯ (1.4) (where also the parametersσα,β,bα,β, cα,β,jα,β are varied) as well as some integro-PDEs arising in option pricing theory in financial markets driven by L´evy processes. To our knowledge, explicit continuous dependence estimates like (1.7) have not appeared in the literature before. Moreover, compared to our previous work [33, 34], the results obtained herein are new even in the pure PDE case, since
we allow for growth in the solutions and hence our results can be applied to the PDEs (and integro-PDEs) arising in finance applications. We will come back to a finance application of our results in the last section of this paper.
Let us mention that continuous dependence estimates are relevant when it comes to determining the regularity of viscosity solutions and obtaining explicit error estimates for approximate solutions. We will provide examples of both aspects.
In particular, we derive error estimates for the vanishing viscosity and vanishing jump viscosity methods for the Bellman/Isaacs equation (1.4) as well as for another singular perturbation problem studied first in [39, 36] in a simpler context. The case of numerical methods is more difficult and some of the first results in that direction for the pure PDE version of the convex Bellman equation can be found in [7, 8, 32, 38]. We anticipate that the continuous dependence estimates herein, together with the ideas in [7, 8, 32], can be used to derive error estimates for the Bellman equation of controlled jump-diffusion processes. We intend to investigate this in a future paper. Although we do not pursue this here, let us also mention that estimates like (1.7) may be relevant to the calibration (inverse) problem for finance models based on L´evy processes, e.g., the problem of determining the L´evy densities using, among other things, empirical data.
Let us now put the present paper in a proper perspective regarding previous literature on continuous dependence estimates for viscosity solutions of pure PDEs.
The case of first order time-dependent Hamilton-Jacobi equations is treated in [50]. For second order PDEs, an applications of the comparison principle [18]
gives a useful continuous dependence estimate when, for example, ¯F is of the form F¯ = F +h for some function h = h(x). In general, the estimate provided by the comparison principle is limited in the sense that it cannot, for example, be used to obtain a convergence rate for the vanishing viscosity method. Continuous dependence estimates for degenerate parabolic equations that imply, among other things, a rate of convergence for the vanishing viscosity method have appeared recently in [16] (see also [30]) and [33, 34]. In particular, [33] and [34] contain results that are general enough to include the Bellman equation associated with optimal control of degenerate diffusion processes as well as the Isaacs equation of zero-sum two-player stochastic differential games. Recently a modification of our continuous dependence estimate in [33], accounting for sub-quadratic growing solutions, was used as one key step in the proof in [14] of thex-H¨older regularity of the gradient of solutions to fully nonlinear uniformly parabolic equations.
This paper is organized as follows: Section 2 is devoted to preliminary material related to viscosity solutions and in particular the statement of an “Ishii Lemma”
for parabolic integro-PDEs (the elliptic version was proved recently in [35]). In Section 3 we state and prove our general continuous dependence theorem, which is applied to the Bellman/Isaacs equation (with bounded as well as unbounded viscosity solutions) in Section 4. In Section 5 we present several applications to the Bellman/Isaacs equation that include, among other things, regularity results and error estimates for some singular perturbation problems. Finally, in Section 6 we illustrate our results on some integro-PDEs for pricing European/American options in an incomplete geometric L´evy stock market.
Notations. We end this introduction by collecting some notations that will be used throughout this paper. Ifx, ybelong to an ordered set, then we letx∨yandx∧y denote max(x, y) and min(x, y) respectively. If x belong to U ⊂ Rn and r > 0,
then B(x, r) denotes the ball {x ∈ U : |x| < r}. We use the notation 1U for the function that is 1 in U and 0 outside. By a modulus ω, we mean a positive, nondecreasing, continuous, sub-additive function which is zero at the origin. In the space of symmetric matricesSN we denote by≤the usual ordering (i.e., X ∈SN, 0 ≤ X means that X positive semidefinite) and by | · | the spectral radius norm (i.e., the maximum of the absolute values of the eigenvalues).
Letν be a signed measure. We denote byν+ and ν− its positive and negative part, so thatν=ν+−ν− (the Jordan decomposition). The absolute value or total variation ofν is|ν|=ν++ν−. If ν1 and ν2 are positive measures, we may define the maximum as follows,
ν1∨ν2:=
dν1
d(ν1+ν2)∨ dν2
d(ν1+ν2)
(ν1+ν2),
where the derivatives are Radon-Nikodym derivatives. If there are functionsf1, f2 and a measureν such thatνi=fiν fori= 1,2 thenν1∨ν2= (f1∨f2)ν.
Let Cn(Ω) n = 0,1,2 denote the spaces of n times continuously differentiable functions on Ω, and letC1,2((0, T)×Ω) denote the space of once in time and twice in space Ω continuously differentiable functions. We let U SC(Ω) and LSC(Ω) denote the spaces of upper and lower semicontinuous functions on Ω, andSC(Ω) = U SC(Ω)∪LSC(Ω). A lower index p denotes the polynomial growth at infinity, so Cpn(Ω), Cp1,2((0, T)×Ω), U SCp(Ω), LSCp(Ω), SCp(Ω) consist of functions f from Cn(Ω), C1,2((0, T)×Ω),U SC(Ω), LSC(Ω),SC(Ω), respectively, satisfying the growth condition
|f(x)| ≤C(1 +|x|)p for allx∈Ω (uniformly in tiff depends on time).
Associated to these spaces are weightedL∞norms which we define as follows:
|f|0,r = sup
x∈Ω
|f(x)|
(1 +|x|)r and |g|0,r = sup
t∈(0,T)
|g(t,·)|0,r
for every r ∈R and every locally bounded function f on Ω and g on (0, T)×Ω.
Finally, we let| · |0=| · |0,0.
2. Viscosity solution theory for integro-PDEs.
In this section we provide some background material for viscosity solutions of integro-PDEs that will be needed in the preceding sections. The class of equations that we cover contains both second order PDEs and up to order two integro op- erators. This generality has been considered earlier by [6, 43] using directly the
“maximum principle for semicontinuous functions” [17]. However, although this approach yields the correct results, it has not been justified in general (see [35]).
In [35], the authors justify a slightly different approach which uses a suitably adapted non-local “maximum principle for semicontinuous functions” or Ishii’s Lemma, see Theorem 2.2 below. Here, we will use the abstract formulation given in [35] to derive continuous dependence estimates for (1.1).
For everyt∈[0, T], x, y∈RN, r, s∈R, X, Y ∈SN, andφ, φk, ψ∈Cp1,2(QT) we will use the following assumptions on (1.1):
The function (t, x, r, q, X)7→F(t, x, r, q, X, φ(t,·)) is continuous, and if (C1)
(tk, xk)→(t, x), Dnφk→Dnφlocally uniformly inQT forn= 0,1,2, and|φk(t, x)| ≤C(1 +|x|)p (Cindependent ofkand (t, x)), then
F(tk, xk, r, q, X, φk(tk,·))→F(t, x, r, q, X, φ(t,·)).
IfX ≤Y and (φ−ψ)(t,·) has a global maximum atx, then (C2)
F(t, x, r, q, X, φ(t,·))≥F(t, x, r, q, Y, ψ(t,·)).
There is aγ∈R(independent ofr, s, t, x, q, X, φ) such that ifr≤s, then (C3)
γ(r−s)≤F(t, x, r, q, X, φ(t,·))−F(t, x, s, q, X, φ(t,·)).
For every constantC∈R, (C4)
F(t, x, r, q, X, φ(t,·) +C) =F(t, x, r, q, X, φ(t,·)).
Remark 2.1. The constantsγin (C3) can be assumed to be non-negative. This can be seen by performing an exponential in time scaling of the solution of (1.1).
Definition 2.1 (Test functions). v ∈ U SCp(QT) (v ∈ LSCp(QT)) is a viscos- ity subsolution (viscosity supersolution) of (1.1) if for every (t, x) ∈ QT and φ ∈ Cp1,2(QT) such that (t, x) is a global maximizer (global minimizer) for v−φ,
φt(t, x) +F(t, x, v(t, x), Dφ(t, x), D2φ(t, x), φ(t,·))≤0 (≥0).
We say thatvis aviscosity solutionof (1.1) ifvis both a sub- and supersolution of (1.1).
Note that viscosity solutions according to this definition are continuous, and that this concept of solutions is an extension of classical solutions. Furthermore, without changing the (sub/super) solutions, we may in this definition assume strict maxima and thatu=φat the maximum. See [35] for simple proofs of these statements and more remarks on this abstract formulation.
Next we introduce an alternative definition of viscosity solutions that is needed for proving comparison and uniqueness results. For everyκ∈(0,1), assume that we have a function
Fκ:QT×R×RN ×SN ×SCp(QT)×C1,2(QT)→R
satisfying the following list of assumptions for every t ∈ [0, T], x, y ∈ RN, r, s ∈ R, q ∈ RN, X, Y ∈ SN, u,−v ∈ U SCp(QT), w ∈ SCp(QT), and φ, φk, ψ, ψk ∈ Cp1,2(QT):
Fκ(t, x, φ(t, x), Dφ(t, x), D2φ(t, x), φ(t,·), φ(t,·)) (F0)
=F(t, x, φ(t, x), Dφ(t, x), D2φ(t, x), φ(t,·)).
The function F in (F0) satisfy (C1).
(F1)
IfX ≤Y and both (u−v)(t,·) and (φ−ψ)(t,·) have global maxima at x, (F2)
then Fκ(t, x, r, q, X, u(t,·), φ(t,·))≥Fκ(t, x, r, q, Y, v(t,·), ψ(t,·)).
The function F in (F0) satisfy (C3).
(F3)
For all constantsC1, C2∈R, (F4)
Fκ(t, x, r, q, X, w(t,·) +C1, φ(t,·) +C2) =Fκ(t, x, r, q, X, w(t,·), φ(t,·)).
Ifψk(t,·)→w(t,·) a.e. inRN and|ψk(t, x)| ≤C(1 +|x|p), then (F5)
Fκ(t, x, r, q, X, ψk(t,·), φ(t,·))→Fκ(t, x, r, q, X, u(t,·), φ(t,·)).
Remark 2.2. If (F0) – (F4) hold, then (C1) – (C4) also hold.
Lemma 2.1 (Alternative definition). Assume there exists Fκ satisfying (F0) – (F2), (F4), and (F5) for every κ∈(0,1). Then v ∈U SCp(QT) (v∈LSCp(QT)) is a viscosity subsolution (viscosity supersolution) of (1.1) if and only if for ev- ery (t, x) ∈ QT and φ ∈ C1,2(QT) such that (t, x) is a global maximizer (global minimizer) forv−φ, and for every κ∈(0,1),
φt(t, x) +Fκ(t, x, v(t, x), Dφ(t, x), D2φ(t, x), v(t,·), φ(t,·))≤0 (≥0).
The proof is similar to that in Sayah [45], see also [6, 35]. The next theorem replaces the maximum principle for semicontinuos functions (cf. [17, 18]) when working with integro-PDEs.
Theorem 2.2. Letu,−v∈U SCp(QT),u(t, x),−v(t, x)≤C(1 +|x|2), solve in the viscosity sense
ut+F(t, x, u, Du, D2u, u(·))≤0 and vt+G(t, x, v, Dv, D2v, v(·))≥0, whereF andGsatisfies (C1) – (C4). Letφ∈C1,2((0, T)×RN×RN)and(¯t,x,¯ y)¯ ∈ (0, T)×RN×RN be such that
u(t, x)−v(t, y)−φ(t, x, y)
has a global maximum at (¯t,x,¯ y). Furthermore assume that in a neighborhood of¯ (¯t,x,¯ y)¯ there are continuous functions g0: [0, T]×R2N →R, g1, g2:QT →SN with g0(¯t,x,¯ y)¯ >0, satisfying
D2φ≤g0(t, x, y)
I −I
−I I
+
g1(t, x) 0 0 g2(t, y)
.
If in addition for everyκ∈(0,1) there existFκ andGκ satisfying (F0) – (F5), then for any ¯γ∈(0,12)there are a, b∈RandX, Y ∈SN satisfying
a−b=φt(¯t,x,¯ y)¯ and
−g0(¯t,x,¯ y)¯
¯ γ
I 0 0 I
≤
X 0 0 −Y
−
g1(¯t,x)¯ 0 0 g2(¯t,y)¯
≤g0(¯t,x,¯ y)¯ 1−2¯γ
I −I
−I I (2.1)
such that
a+Fκ(¯t,x, u(¯¯ t,x), D¯ xφ(¯t,x,¯ y), X, u(¯¯ t,·), φ(¯t,·,y))¯ ≤0 and (2.2)
b+Gκ(¯t,y, v(¯¯ y),−Dyφ(¯t,x,¯ y), Y, v(¯¯ t,·),−φ(¯t,x,¯ ·))≥0.
(2.3)
Outline of proof. The theorem is essentially a special case of the corresponding elliptic result Theorem 4.8 in [35]. This follows from the procedure of Section 3 in Crandall and Ishii [17] that we will repeat here for the readers’ convenience.
We may assume that the maximum is strict. Then the function u(t, x)−v(s, y)−φ(t, x, y)−1
δ(t−s)2,
will have a global maximum at some point (˜t,˜s,x,˜ y)˜ ∈[0, T]2×R2N. Furthermore, asδ→0, along a subsequence (˜t,˜s,x,˜ y)˜ →(¯t,¯t,x,¯ y) and¯ 1δ(˜t−˜s)2→0. Choosing δsmall enough, we have (˜t,s)˜ ∈(0, T)2. Lettingψ(t, s, x, y) :=φ(t, x, y) +1δ(t−s)2, it is not difficult to see that
ψt−ψs=φt and Dnψ=Dnφ (n= 1,2).
With this in mind, we apply the elliptic result (Lemma 7.8) in [35]. The result is the existence of two matrices ˜X,Y˜ ∈SN satisfying
−g0(˜t,x,˜ y)˜
¯ γ
I 0 0 I
≤
X˜ 0 0 −Y˜
−
g1(˜t,x)˜ 0 0 g2(˜t,y)˜
≤g0(˜t,x,˜ y)˜ 1−2¯γ
I −I
−I I
such that
˜
a+F(˜t,x, u(˜˜ t,x), D˜ xφ(˜t,x,˜ y),˜ X, φ(˜˜ t,·,y))˜ ≤0 and
˜b+G(˜s,y, v(˜˜ y),−Dyφ(˜t,x,˜ y),˜ Y ,˜ −φ(˜t,x,˜ ·))≥0,
where ˜a := ψt(˜t,s,˜ x,˜ y) and ˜˜ b := ψs(˜t,s,˜ x,˜ y). Observe that we use the˜ F/G- formulation, and not theFκ/Gκ-formulation at this point. Also note that by (C4) the (t−s)2part inψdoes not appear in the non-local part in the above inequalities because it is a constant w.r.t.xandy.
The inequalities give upper bounds on ˜aand−˜b, and since ˜a−˜b=φt(˜t,x,˜ y), the˜ two sequences are bounded inδ. We may therefore extract converging subsequences of ˜a,˜b,X,˜ Y˜ as δ → 0. Denoting the limits by a, b, X, Y, we obtain the result by sending δ → 0 along this subsequence, using (semi) continuity of all involved functions.
The final step is to show that a similar result holds in theFκ/Gκ formulation.
We omit this easy step and refer the interested reader to the proof of Theorem 4.8
in [35], see also Lemma 2.1 above.
Remark 2.3. The technical conditionu(x),−v(x)≤C(1 +|x|2) is an artifact of the method used to prove Theorem 4.8 in [35]. It does not seem easy to remove. In practice, however, it creates no difficulties.
Remark 2.4. Using the notation of [18], we note that
(a, Dxφ(¯t,x,¯ y), X)¯ ∈ J2,+u(¯t,x)¯ and (b,−Dyφ(¯t,x,¯ y), Y¯ )∈ J2,−v(¯t,y).¯ But as opposed the pure PDE case, a priori we do not know that the viscosity inequalities hold for elements in J2,+u(¯t,x) and¯ J2,−v(¯t,y) respectively, see [35]¯ for a discussion of this point in the elliptic setting.
3. Continuous Dependence Estimates.
In this section we formulate and prove an abstract continuous dependence es- timate for Integro-PDEs. It is a pointwise estimate which may have polynomial growth in the space variable x. As will be explained in the following, this result is an extension of results in [34] (see also [33, 16]) in two directions: (i) We have equations with an integro operator and (ii) we allow for (polynomial) growth in the estimates. In the next sections we will see how this rather complicated and abstract result can be used to obtain new continuous dependence estimates for the Bellman/Isaacs and Black-Scholes type equations.
The following crucial condition can be thought of as a “continuous dependence”
version of condition (3.14) in the User’s Guide [18]. For every κ ∈ (0,1), t ∈ [0, T], x, y ∈ RN, r, s ∈ R, q ∈ RN, X, Y ∈ SN, u,−v ∈ U SCm(QT), and φ ∈ C1,2(QT) we assume:
Letα, ε, λ >0,p≥2, and define (F6)
φ(t, x, y) =eλtα
2|x−y|2+eλtε
p(|x|p+|y|p).
There are constantsη1, . . . , η4, p1, . . . , p4, ps, K1, K2, K3≥0 independent ofα, ε, λ, t, and a modulusmα,ε(depending on α, ε) such that
wheneveru(t, x)−v(t, y)−φ(t, x, y) has a global maximum at (¯t,x,¯ y),¯ F¯κ
¯t,y, r, e¯ λ¯tα(¯x−y)¯ −eλ¯tε¯y|¯y|p−2, Y, v(¯t,·),−φ(¯t,x,¯ ·)
−Fκ
¯t,x, r, e¯ λ¯tα(¯x−y) +¯ eλ¯tε¯x|x|¯p−2, X, u(¯t,·), φ(¯t,·,y)¯
≤
2
X
i=1
(1 +|¯x|+|¯y|)piηi+α
4
X
i=3
(1 +|¯x|+|¯y|)2piη2i +K1(1 +|¯x|+|¯y|)ps|¯x−y|¯ +K2eλt¯α|¯x−y|¯2 +K3eλ¯tε(1 +|¯x|p+|¯y|p) +mα,ε(κ),
for every|r| ≤ |u|0∧ |¯u|0, and X, Y satisfying X 0
0 −Y
≤2eλ¯tα
I −I
−I I
+eλt¯ε(p−1)
|¯x|p−2I 0 0 |¯y|p−2I
. (3.1)
The matrix inequality above corresponds to the second inequality in (2.1) when
¯
γ= 1/4 andφis as defined above.
Theorem 3.1(Continuous Dependence Estimate). Let p≥2 andm < p, let F,F¯ andFκ,F¯κ, κ∈(0,1) be functions satisfying assumptions (C1) – (C4) and (F0) – (F6) respectively, and let u,−¯u∈U SCm(QT)satisfy in the viscosity sense
ut(t, x) +F(t, x, u(t, x), Du(t, x), D2u(t, x), u(t,·))≤0 and
¯
ut(t, x) + ¯F(t, x,u(t, x), D¯ u(t, x), D¯ 2u(t, x),¯ u(t,¯ ·))≥0.
Furthermore, letp0≥0 (p0 is used in (3.2)), assume (F6) holds with p >2 max(p0, . . . , p4, ps),
and assume
|Du(0, x)|,|D¯u(0, x)| ≤K4(1 +|x|+|y|)ps a.e.
Then there is a constantC >0(depending only onK1, . . . , K4, p0, . . . , p4, ps, p, T) such that for every(t, x)∈QT:
u(t, x)−¯u(t, x)≤C(1 +|x|)p0
(u(0,·)−u(0,¯ ·))+ 0,p
0
+C
2
X
i=1
T1−pip(1 +|x|)piηi+C
4
X
i=3
T12−pip(1 +|x|)pi+psηi. (3.2)
Before giving the proof we give some remarks and corollaries.
Remark 3.1. We have not specified the various constants in Theorem 3.1, but it is possible to get bounds on them by tracing them in the proof below. However, getting optimal bounds would be difficult from the present proof because of the complexity, all the approximations used, and arbitrariness of the form that one factor/term can be decreased at the expense of increasing another factor/term.
However, if all the constantsp’s andK’s are independent ofT, it follows from the proof that the various constantsCcan be chosen to be positive, finite, continuous in T, and strictly positive in the limit asT →0. In addition, it follows that whenever one of the exponents p0, p1, p2 is equal to 0, we may take the corresponding C in (3.2) to be 1.
Let us now consider a special case where uand ¯uare bounded and there is no growth in the data, i.e.,m=p0=· · ·=p4=ps= 0.
Corollary 3.2. Assume that the assumptions of Theorem 3.1 are satisfied with m=p0=· · ·=p4=ps= 0andη2=η4= 0. Then there is a constantC >0such that
|(u−u)¯ +|0≤ |(u(0,·)−u(0,¯ ·))+|0+T η1+CT1/2η3.
This corollary is an extension of Theorem 2.1 in [34] to Integro-PDEs. The coefficient 1 in front of theT η1-term is explained in Remark 3.1. Next we consider the case whereuand ¯uare both continuous. Theorem 3.1 gives an upper bound on
u(t, x)−u(t, x)¯
valid for allt∈[0, T) andx∈RN. Furthermore, this bound is independent oft, so by sendingt→T and using continuity the same bound also holds for
u(T, x)−u(T, x).¯ RenamingT totwe then have the following result:
Corollary 3.3. (a) Assume that the assumptions of Theorem 3.1 hold and in addition that u,¯u∈ C(QT). Then there is a constant C > 0 (independent of t) such that for every(t, x)∈QT,
u(t, x)−u(t, x)¯ ≤C(1 +|x|)p0
(u(0,·)−u(0,¯ ·))+ 0,p
0
+C
2
X
i=1
t1−pip(1 +|x|)piηi+C
4
X
i=3
t12−pip(1 +|x|)pi+psηi.
(b) Assume that the assumptions of Corollary 3.2 hold and in addition that u,u¯∈ C(QT). Then there is a constantC >0(independent of t) such that
u(t, x)−u(t, x)¯ ≤ |(u(0,·)−u(0,¯ ·))+|0+tη1+Ct1/2η3.
That fact that the constants C can be chosen independently of t follows from Remark 3.1. Take as new constants the maximum over [0, T] of the t-depending C’s given by Theorem 3.1.
Remark 3.2. Notice the time dependence in the estimate in Corollary 3.3 (a). It differs from the time dependency in Corollary 3.3 (b) when pi > 0 for at least one i ∈ {1,2,3,4}. This is an effect of the growth in the data (and hence in the solutions).
In the above bounds on u−u,¯ p behaves like a free parameter. It may vary between its lower bound and any number p for which the non-local part of the equation is well-defined (so no restrictions for pure PDEs!). If we were allowed to sendp→ ∞, we would obtain theT-exponents (t-exponents) 1 and 1/2. However, our estimates do not allow this, since the way we do the proof, at least some of the constantsC will blow up asp→ ∞.
Remark 3.3. The complicated condition (F6) is a natural “structure condition”
leading to continuous dependence estimates in the viscosity solutions setting. The use of this condition will be clearer in the next section where we derive both known and new continuous dependence results for Bellman/Isaacs equations under as- sumptions that include the Black-Scholes equation. The new features here con- sist of estimates on the integro operators and allowing for estimates with growth.
Growth in the estimates arise naturally when studying Black-Scholes type of equa- tions where the underlying stochastic process is an exponential L´evy process. In the following sections, we will present examples where some or all of the exponents p0,· · ·, p4, ps are different from 0.
Finally, we remark that Theorem 3.1 allows for four error termsη1, . . . , η4(with correspondingp1, . . . , p4). In Corollary 3.2 and in [34], only two terms were used.
One could consider any number of such error terms η, both in the above theorem and in applications, but in this paper we confine ourselves to situations where up to four error terms are sufficient.
Now we turn to the proof of Theorem 3.1.
Proof of Theorem 3.1. We may assume that γ ≥0, see Remark 2.1. Let us start by defining the following quantities
ψ(t, x, y) :=u(t, x)−u(t, y)¯ −φ(t, x, y)−δσ T t− ε¯
T−t whereδ,ε¯∈(0,1) and
σ0:= sup
x,y∈RN
n
u(0, x)−u(0, y)¯ −φ(0, x, y)− ε¯ T
o+ ,
σ:= sup
t∈[0,T) x,y∈RN
u(t, x)−u(t, y)¯ −φ(t, x, y)− ε¯ T−t
−σ0.
By the continuity ofψ, precompactness of sets of the type{ψ(t, x, y)> k}, and the penalization term T−tε¯ , there existst0∈[0, T),x0, y0∈RN such that
sup
t∈[0,T),x,y∈RN
ψ(t, x, y) =ψ(t0, x0, y0).
We want an upper bound on σ+σ0, and we start by deriving a positive upper bound forσ. We may therefore assume thatσ >0. This implies that t0>0, since on one hand
ψ(t0, x0, y0)≥σ+σ0−δσ > σ0,
while on the other handt0= 0 would imply ψ(t0, x0, y0)≤σ0, which is a contra- diction.
We can now apply Theorem 2.2 (with ¯γ = 1/4) to conclude that there are numbers a, b ∈ R satisfying a−b = φt(t0, x0, y0) +δσT +(T−t)ε¯ 2, and symmetric matrices X, Y ∈ SN satisfying inequality (3.1) such that the following inequality holds
a−b≤F¯κ(t0, y0,u(t¯ 0, y0),−Dyφ(t0, x0, y0), Y,u(t¯ 0,·),−φ(t0, x0,·))
−Fκ(t0, x0, u(t0, x0), Dxφ(t0, x0, y0), X, u(t0,·), φ(t0,·, y0)).
Sinceσ >0 it follows thatu(t0, x0)≥u(t¯ 0, y0), so after using (F3) with γ≥0, (F6), and the above inequality, we have
δσ T +λ
2eλt0α|x0−y0|2+λ
peλt0ε(|x0|p+|y0|p)
≤
2
X
i=1
(1 +|x0|+|y0|)piηi+α
4
X
i=3
(1 +|x0|+|y0|)2piη2i
+K1(1 +|x0|+|y0|)ps|x0−y0|+K2eλt0α|x0−y0|2 +K3eλt0ε(1 +|x0|p+|y0|p) +mα,ε(κ).
We sendκ→0 and chooseλto satisfy
λ= 2(K2+ 1)∨p(K3+ 1)
(the number +1 is an arbitrarily chosen positive number) and obtain δσ
T ≤
2
X
i=1
(1 +|x0|+|y0|)piηi+α
4
X
i=3
(1 +|x0|+|y0|)2piηi2
+K1(1 +|x0|+|y0|)ps|x0−y0| −eλt0α|x0−y0|2−eλt0ε(1 +|x0|p+|y0|p). Then we sendδ→1, maximize w.r.t. |x0−y0|, and use
3−p+1(1 +|x0|+|y0|)p≤1 +|x0|p+|y0|p to obtain
σ T ≤
2
X
i=1
(1 +|x0|+|y0|)piηi+α
4
X
i=3
(1 +|x0|+|y0|)2piη2i +Cα−1(1 +|x0|+|y0|)2ps−Cε(1 +|x0|+|y0|)p :=
2
X
i=1
Ai(r) +
4
X
i=3
Ai(r) +A5(r)−Cεrp where r= 1 +|x0|+|y0|.
Now letri denote the maximum point of Ai(r)−1
5Cεrp, fori= 1,2, . . . ,5. That is
ri=Cηi
ε p−1pi
, i= 1,2; ri=C αη2i
ε
p−21pi
, i= 3,4; r5=C(εα)−p−2ps1 . Then we have
σ≤T
5
X
i=1
Ai(ri)−1 5Crpi
=CT
2
X
i=1
ε−p−pipiη
p p−pi
i +CT
4
X
i=3
ε−
2pi
p−2pi(αη2i)p−2ppi +CT ε−p−2ps2ps α−p−2psp . Now we need an estimate of σ0. Using the regularity of the initial values and a similar optimization procedure as we used above, we obtain
σ0≤Cε−
p0 p−p0
(u(0,·)−u(0,¯ ·))+ 1 +| · |p0
p p−p0
0
+Cε−p−2ps2ps α−p−2psp . By the calculations above we have
σ+σ0≤Cε−
p0 p−p0M
p p−p0
0 +CT
2
X
i=1
ε−p−pipiη
p p−pi
i
+CT
4
X
i=3
ε−
2pi
p−2pi(αηi2)p−2ppi +Cε−p−2ps2ps α−p−2psp
:=B0+
2
X
i=1
Bi+
4
X
i=3
Bi(α) +B5(α),
where M0 denotes the weighted norm of the initial conditions. Note that this expression holds for all positiveα. We proceed to obtain an upper bound onσ+σ0
that does not depend onαby choosing a suboptimalα. Letα3andα4respectively denote the minimum points of
Bi(α) +B5(α) =CT ε−
2pi
p−2pi(αη2i)p−2ppi +Cε−p−2ps2ps α−p−2psp , fori= 3 andi= 4, i.e.
αi=CT−
(p−2pi)(p−2ps) 2p(p−pi−ps) η−
p−2ps p−pi−ps
i εp−pipi−ps−ps for i= 3,4.
Then set
¯
α= min{α3, α4}
and note that since ¯α ≤ α3 and ¯α ≤ α4, the definitions of ¯α, α3, α4 lead to the following bound
σ+σ0≤B1+
2
X
i=1
Bi+
4
X
i=3
Bi( ¯α) +B5( ¯α) (3.3)
≤B1+
2
X
i=1
Bi+
4
X
i=3
Bi(αi) +B5( ¯α)
=Cε−
p0 p−p0M
p p−p0
0 +CT
2
X
i=1
ε−p−pipiη
p p−pi
i
+C
4
X
i=3
T
p−2pi 2p−2pi−2psη
p p−pi−ps
i ε−p−pipi+ps−ps
:=A0(ε) +
2
X
i=1
Ai(ε) +
4
X
i=3
Ai(ε), which holds for anyε >0.
To complete the proof, we use the definition ofσto see that u(t, x)−u(t, x)¯ −2ε
peλt|x|p− ε¯
T−t ≤σ+σ0
for any (t, x)∈QT. We send ¯ε→0, use|x|p ≤(1 +|x|)p, and use the bound (3.3), to see that is
u(t, x)−u(t, x)¯ ≤σ+σ0+2ε
peλT(1 +|x|)p
≤A0(ε) +
2
X
i=1
Ai(ε) +
4
X
i=3
Ai(ε) +2ε
peλT(1 +|x|)p. This bound holds for everyε >0. Next we find a bound independent ofε. Let εi be the minimum point of
Ai(ε) +εC(1 +|x|)p fori= 0, . . . ,4, i.e.,
ε0=CM0(1 +|x|)p0−p, εi=CTp−ppiηi(1 +|x|)pi−p, i= 1,2,
and
εi =CT
p−2pi
2p ηi(1 +|x|)pi+ps−p, i= 3,4.
Now we set
¯
ε= max(ε1, . . . , ε5).
With this value ofε, since ¯εi≤εfori= 0, . . . ,4, we have u(t, x)−u(t, x)¯ ≤A0(¯ε) +
2
X
i=1
Ai(¯ε) +
4
X
i=3
Ai(¯ε) +2¯ε
peλT(1 +|x|)p
≤A0(ε0) +
2
X
i=1
Ai(εi) +
4
X
i=3
Ai(εi) +2¯ε
peλT(1 +|x|)p,
which is (3.2) and the proof is complete.
4. The Bellman/Isaacs equation
In this section we consider the Bellman/Isaacs equation (1.4) with initial values u(0, x) =u0(x) in RN.
(4.1)
We will state assumptions that are natural and standard in view of the connections to the theory of stochastic control and differential games, see [25, 37, 26, 49, 43].
Under these assumptions we then derive continuous dependence results for sub- and supersolutions that are bounded or have polynomial growth at infinity.
We assume that there are constants K1, . . . , K5, Kt,x ≥ 0, λ∈ R, p≥ 2, and a function ρ ≥ 0 such that the following statements hold for every t ∈ [0, T], x, y∈RN, α∈ A, β∈ B, andz∈RM \ {0}:
σ, b, c, f, jare continuous w.r.t. t, x, α, βand Borel measurable (A0)
w.r.t. z;A,B are compact metric spaces;πis a positiveσ-finite Radon measure onRM\ {0} satisfyingπ({0}) = 0 and
K0:=
Z
B(0,1)\{0}
ρ(z)2π(dz) + Z
RM\B(0,1)
(1 +ρ(z))pπ(dz)<∞,
|fα,β(t, x)−fα,β(t, y)|+|u0(x)−u0(y)| ≤K1|x−y|, (A1)
cα,β ≥λ and |cα,β(t, x)−cα,β(t, y)| ≤K2|x−y|, (A2)
|σα,β(t, x)−σα,β(t, y)|+|bα,β(t, x)−bα,β(t, y)| ≤K3|x−y|, (A3)
|jα,β(t, x, z)| ≤K4ρ(z)(1 +|x|), |jα,β(t, x, z)|χB(0,1)(z)≤Kt,x, (A4)
and|jα,β(t, x, z)−jα,β(t, y, z)| ≤K5ρ(z)|x−y|.
The L´evy measure π(dz) may have a singularity at z = 0. As an example in R1, take ρ(z) = |z| and π(dz) = z−δχB(0,1)(z) where δ ∈ (0,3). Furthermore, it integrates functions growing like (1 +ρ(x))p at infinity. If assumptions (A0) and (A4) hold, then the integral part of the Bellman/Isaacs equation (1.4) is well defined for functions inCp1,2(QT), see, e.g., Garroni and Menaldi [29]. Assumptions (A0) – (A4) were used in Pham [43] to obtain comparison results for second order integro-PDEs, see also [35].
Note that (A1) – (A3) imply
|σα,β(t, x)|+|bα,β(t, x)|+|cα,β(t, x)|+|fα,β(t, x)|+|u0(x)| ≤C(1 +|x|),
for some constant C >0. The growth at infinity of the solutions of (1.4) is equal to the growth of the fastest growing function among the initial data u0 and the
“source term”f. Hence, assumption (A1) leads to at most linear growth.
We will now state the continuous dependence results. Fori= 1,2 we consider a sub- or supersolutionui of
uit+ inf
α∈Asup
β∈B
n−Lα,βi ui− Bα,βi ui+fiα,βo
= 0 in QT, ui(0, x) =ui0(x) in RN, (4.2)
whereLα,βi andBα,βi are the operators defined in (1.5) corresponding toσi, bi, ci, ji, πi. Theorem 4.1(Bounded Case I). Assumeσi,bi,ci,fi,ui0,ji,πi,i= 1,2, satisfy (A0) – (A4), u1 ∈U SC0(QT) is a viscosity subsolution of (4.2) with i = 1, and u2∈LSC0(QT)is a viscosity supersolution of (4.2)withi= 2. Then the following pointwise estimate holds:
u1(t, x)−u2(t, x)≤
(u10−u20)+ 0 +Tsup
α,β
|f1−f2|0+|u1|0∨ |u2|0|c1−c2|0 +CT1/2
sup
α,β
|σ1−σ2|0+ sup
α,β
|b1−b2|0
+CT1/2sup
α,β
Z
RN\{0}
|j1−j2|2π(dz)
1/2 0
+CT1/2(1 +|x|) sup
α,β
Z
RN\{0}
j2|π1−π2|(dz)
1/2 0,2, whereπ= max{π1, π2}andj = max{j1, j2}.
We can get better results whenu1 andu2 are more regular. We will only state one such result.
Theorem 4.2(Bounded Case II). Assumeσi,bi,ci,fi,ui0,ji,πi,i= 1,2, satisfy (A0) – (A4),u1∈C(QT)is a viscosity subsolution of (4.2)withi= 1,u2∈C(QT) is a viscosity supersolution of (4.2)with i= 2, and
|Du1|0+|Du2|0<∞.
Then the following pointwise estimate holds:
u1(t, x)−u2(t, x)≤
(u10−u20)+ 0 +tsup
α,β
|f1−f2|0+|u1|0∨ |u2|0|c1−c2|0+|Du1|0∨ |Du2|0|b1−b2|0
+Ct1/2sup
α,β
(
|σ1−σ2|0+ Z
RN\{0}
|j1−j2|2π(dz)
1/2 0
)
+Ct1/2(1 +|x|) sup
α,β
Z
RN\{0}
j2|π1−π2|(dz)
1/2 0,2
, whereπ= max{π1, π2}andj = max{j1, j2}.
In the case of sub- and supersolutions with polynomial growth, we will relax assumption (A1) and strengthen assumption (A2) in the following way:
There is a real number ps≥0 such that (A1’)
|fα,β(t, x)−fα,β(t, y)|+|u0(x)−u0(y)| ≤K1(1 +|x|+|y|)ps|x−y|.
cα,β≥λ and cα,β is constant for eachα∈ Aandβ ∈ B.
(A2’)
These assumptions have been used in Krylov [37] (but see Remark 4.2), where the convex Bellman equation without an integro operator is considered. See also [25].
Note that (A1’) implies the following bound onf andu0
|fα,β(t, x)|+|u0(x)| ≤C(1 +|x|)1+ps.
In view of earlier remarks, such a bound also applies to the solutions of (1.4). In particular, ifps= 0, the solutions have (at most) linear growth at infinity.
Theorem 4.3 (Polynomial growth). Assume σi, bi, ci, fi, ui0, ji, πi, i = 1,2, satisfy (A0), (A1’), (A2’), (A3), and (A4), u1 ∈ U SC1+ps(QT) is a viscosity subsolution of (4.2)withi= 1, and u2∈LSC1+ps(QT)is a viscosity supersolution of (4.2) with i = 2. Let R, r ≥ 0. If p > 2 max(R, r,1 +ps), then the following pointwise estimate holds:
u1(t, x)−u2(t, x)
≤C(1 +|x|)R
(u10−u20)+
0,R+T1−Rp sup
α,β
|f1−f2|0,R +CT1−1+psp (1 +|x|)1+pssup
α,β
|c1−c2| +CT12−rp(1 +|x|)r+ps
×sup
α,β
|σ1−σ2|0,r+|b1−b2|0,r+ Z
RN\{0}
|j1−j2|2π(dz)
1/2 0,r
+CT12−1p(1 +|x|)1+pssup
α,β
Z
RN\{0}
j2|π1−π2|(dz)
1/2 0,2, whereπ= max{π1, π2}andj = max{j1, j2}.
Remark 4.1. The various constantsC in the above two theorems depend on inte- grability and Lipschitz bounds and growth at infinity of the data/initial values of two problems, and also on the constant λ defined in (A2)/(A2’). In other words, the various constants and exponents defined in (A0) – (A4), (A1’), and (A2’).
We also remark that all constantsCin the two theorems above, except the ones in front of the|π1−π2|terms, can be chosen to depend only on one of the data-sets.
Either theu1-data or theu2-data. This fact is written out explicitly in [33].
In applications, the constantsRandrappearing in Theorem 4.3 are to be chosen such that the weighted norms are finite. In the next section, we will see examples where (i)R=psandr= 0 and (ii)R= 1 +ps andr= 1. Note that one could let all the weighted norms above be different (have differentR’s andr’s), but we have omitted this case for simplicity.
Remark 4.2. The restrictive assumption (A2’) was introduced to simplify the es- timates. With this assumption the structure of the equation is respected in the sense that the coefficients of the i-th order term isO(xi) fori= 0,1,2. We could, however, use a more general assumption like the following used by Krylov [37]:
|cα,β(t, x)−cα,β(t, y)| ≤K2(1 +|x|+|y|)pc|x−y|
for somepc≥0. In addition to modifications to thec-term, the effect on Theorem 4.3 would be to replaceps by 1 +ps+pc in the last two terms.
Remark 4.3. Due to the complexity of the problems considered here, it is not possible to give one continuous dependence result that is well suited for every special case. We have given some results that are good for problems with order two integro operators and the specified regularity of the sub- and supersolutions. By varying the assumptions, many other (mostly easier) results can be obtained from Theorem 3.1. Let us mention a few possible modifications:
• Better estimates can be had for integral operators of order 0 and 1, at least when the solutions are, e.g., Lipschitz continuous.
• Estimates for locally H¨older continuousu0, f, ccan be obtained by adapting the arguments in [33] for the global H¨older case.
• When jump-vectorsj1, j2arex-bounded, the estimate of Theorems 4.1 and 4.2 have no growth.
Proofs of Theorems 4.1 – 4.3. The theorems will be proved by invoking Theorem 3.1 (see also Remark 3.1 and Corollary 3.3), so we have to define the appropriate functions F, Fκ and check that they satisfy assumptions (C1) – (C4) and (F0) – (F6). We set
F(t, x, r, q, X, φ(t,·))
= inf
β∈Bsup
α∈A
n−tr[aα,β(t, x)X]−bα,β(t, x)q+cα,β(t, x)r+fα,β(t, x)
− Z
RM\{0}
φ(t, x+jα,β(t, x, z))−φ(t, x)−jα,β(t, x, z)q π(dz)o and
Fκ(t, x, r, q, X, v(t,·), φ(t,·))
= inf
β∈Bsup
α∈A
n−tr[aα,β(t, x)X]−bα,β(t, x)q+cα,β(t, x)r +fα,β(t, x)−Bκα,β(t, x, q, φ(t,·))−Bα,β,κ(t, x, q, v(t,·))o whereaα,β is defined in (1.5) and
Bα,βκ (t, x, q, φ(t,·))
= Z
B(0,κ)\{0}
φ(t, x+jα,β(t, x, z))−φ(t, x)−jα,β(t, x, z)q π(dz), Bα,β,κ(t, x, q, v(t,·))
= Z
RM\B(0,κ)
v(t, x+jα,β(t, x, z))−v(t, x)−jα,β(t, x, z)q π(dz).
Note thatpis defined in (A0). By (A0) and (A4),F satisfies (C1) – (C4) and Fκ
satisfies (F0) – (F5).
The main difficulty is assumption (F6). For Theorem 4.1 to be true, the constants in (F6) must be the following: p1=p2=p3= 0,p4= 1,ps= 0,
η1= sup
α,β
|f1−f2|0+|u1|0∨ |u2|0|c1−c2|0 , η2= 0,
η32=Csup
α,β
n|σ1−σ2|20+|b1−b2|20+ Z
RN\{0}
|j1−j2|2π(dz) 0
o ,
